In
mathematics, the HNN extension is an important construction of
combinatorial group theory In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a nat ...
.
Introduced in a 1949 paper ''Embedding Theorems for Groups'' by
Graham Higman
Graham Higman FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory.
Biography
Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winning a ...
,
Bernhard Neumann
Bernhard Hermann Neumann (15 October 1909 – 21 October 2002) was a German-born British-Australian mathematician, who was a leader in the study of group theory.
Early life and education
After gaining a D.Phil. from Friedrich-Wilhelms Universit ...
, and
Hanna Neumann
Johanna (Hanna) Neumann (née von Caemmerer; 12 February 1914 – 14 November 1971) was a German-born mathematician who worked on group theory.
Biography
Neumann was born on 12 February 1914 in Lankwitz, Steglitz-Zehlendorf (today a distr ...
, it embeds a given group ''G'' into another group ''G' '', in such a way that two given isomorphic subgroups of ''G'' are conjugate (through a given isomorphism) in ''G' ''.
Construction
Let ''G'' be a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
with
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
, and let
be an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
between two subgroups of ''G''. Let ''t'' be a new symbol not in ''S'', and define
:
The group
is called the ''HNN extension of'' ''G'' ''relative to'' α. The original group G is called the ''base group'' for the construction, while the subgroups ''H'' and ''K'' are the ''associated subgroups''. The new generator ''t'' is called the ''stable letter''.
Key properties
Since the presentation for
contains all the generators and relations from the presentation for ''G'', there is a natural homomorphism, induced by the identification of generators, which takes ''G'' to
. Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of ''G'' into
. A consequence is that two isomorphic subgroups of a given group are always conjugate in some
overgroup; the desire to show this was the original motivation for the construction.
Britton's Lemma
A key property of HNN-extensions is a normal form theorem known as Britton's Lemma. Let
be as above and let ''w'' be the following product in
:
:
Then Britton's Lemma can be stated as follows:
Britton's Lemma. If ''w'' = 1 in ''G''∗α then
*either and ''g''0 = 1 in ''G''
*or and for some ''i'' ∈ one of the following holds:
#ε''i'' = 1, ε''i''+1 = −1, ''gi'' ∈ ''H'',
#ε''i'' = −1, ε''i''+1 = 1, ''gi'' ∈ ''K''.
In contrapositive terms, Britton's Lemma takes the following form:
Britton's Lemma (alternate form). If ''w'' is such that
*either and ''g''0 ≠ 1 ∈ ''G'',
*or and the product ''w'' does not contain substrings of the form ''tht''−1, where ''h'' ∈ ''H'' and of the form ''t''−1''kt'' where ''k'' ∈ ''K'',
then in .
Consequences of Britton's Lemma
Most basic properties of HNN-extensions follow from Britton's Lemma. These consequences include the following facts:
*The natural
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from ''G'' to
is injective, so that we can think of
as containing ''G'' as a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
.
*Every element of finite order in
is
conjugate to an element of ''G''.
*Every finite subgroup of
is conjugate to a finite subgroup of ''G''.
*If
contains an element
such that
is contained in neither
nor
for any integer
, then
contains a subgroup isomorphic to a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
of rank two.
Applications and generalizations
Applied to
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the HNN extension constructs the
fundamental group of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' that has been 'glued back' on itself by a mapping ''f : X → X'' (see e.g.
Surface bundle over the circle
In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of mappi ...
). Thus, HNN extensions describe the fundamental group of a self-glued space in the same way that
free products with amalgamation do for two spaces ''X'' and ''Y'' glued along a connected common subspace, as in the
Seifert-van Kampen theorem. These two constructions allow the description of the fundamental group of any reasonable geometric gluing. This is generalized into the
Bass–Serre theory Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as ...
of groups acting on trees, constructing fundamental groups of
graphs of groups.
HNN-extensions play a key role in Higman's proof of the
Higman embedding theorem which states that every
finitely generated recursively presented group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
can be homomorphically embedded in a
finitely presented group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
. Most modern proofs of the
Novikov–Boone theorem
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
about the existence of a
finitely presented group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
with algorithmically undecidable
word problem also substantially use HNN-extensions.
The idea of HNN extension has been extended to other parts of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, including
Lie algebra theory.
References
{{reflist
Group theory
Combinatorics on words